Precise polynomial heuristic for an NP-complete problem

TL;DR

This paper presents a new polynomial heuristic for the NP-complete minimum vertex cover problem, which accurately finds solutions for various graph types up to large sizes by using an iterative probability-based approach.

Contribution

The authors introduce a simple, efficient, and precise polynomial heuristic that iteratively finds stable probability solutions and derives exact minimum vertex covers, extending applicability to large graphs.

Findings

Correctly solves all tested small graphs for minimum vertex cover

Provides precise data for graphs up to 50,000 sites

Discusses potential extensions to other NP problems

Abstract

We introduce a simple, efficient and precise polynomial heuristic for a key NP complete problem, minimum vertex cover. Our method is iterative and operates in probability space. Once a stable probability solution is found we find the true combinatorial solution from the probabilities. For system sizes which are amenable to exact solution by conventional means, we find a correct minimum vertex cover for all cases which we have tested, which include random graphs and diluted triangular lattices of up to 100 sites. We present precise data for minimum vertex cover on graphs of up to 50,000 sites. Extensions of the method to hard core lattices gases and other NP problems are discussed.